Example

The straightening functor effectively computes the fibers of a Cartesian fibration(p:X→C)(p : X \to C) over every point x∈Cx \in C. As an illustration for how this is expressed in terms of morphisms in that pushout, consider the simple situation where

C=*C = * only has a single point;

X={a→bc}X = \left\{ a \to b \;\;\; c\right\} is a category with three objects, two of them connected by a morphism

p:X→Cp : X\to C is the only possible functor, sending everything to the point.

(because K(ϕ,p)K(\phi,p) is CC with a single object ν\nu and some morphisms to ν\nu adjoined, such that there are no non-degenerate morphisms originating at ν\nu, we have that K(ϕ,p)K(\phi,p) is of form CFC_F for some FF; and Stϕ(X)St_\phi(X) is that FF by definition).

Using the pasting law for pushouts (see pullback) we just have to compute the lower square pushout. Here the statement is a special case of the following statement: for every sSet-category of the form CFC_F, the pushout of the canonical inclusion C→CFC\to C_F along any sSetsSet-functor π:C→C′\pi : C \to C' is C′π!FC'_{\pi_! F}.

Now by the definition of left Kan extensionπ!\pi_! as the left adjoint to prescomposition with a functor, this is bijectively a transformation

η:π!F→r*Q(−,d(ν)).
\eta : \pi_! F \to r^* Q(-,d(\nu))
\,.

Using this we see that we may find a universal cocone by setting Q:=C′π!FQ := C'_{\pi_! F} with r:C′→Qr : C' \to Q the canonical inclusion and CF→C′π!FC_{F} \to C'_{\pi_! F} given by π\pi on the restriction to CC and by the unitF→π*π!FF \to \pi^* \pi_! F on CF(c,ν)C_F(c,\nu). For this the adjunct transformation η\eta is the identity, which makes this universal among all cocones.

where the top morphism is an equivalence and the right morphism a Kan fibration. Moreover, as discussed at right fibration, over an ∞\infty-groupoid the notions of left/right fibrations and Kan fibrations coincide. This shows that the full sub-(∞,1)-category of ∞Grpd/X\infty Grpd/X on the right fibrations is equivalent to all of ∞Grpd/X\infty Grpd/X.

We define the straightening functor to assign that marking of edges which is the minimal one such that all such morphisms f˜\tilde f are marked in StϕX(d)St_\phi X(d), for all marked f:d→ef : d \to e in XX: this means that this marking is being completed under the constraint that Stϕ(X)St_\phi(X) be sSet-enriched functorial.

For that, recall that the hom simplicial sets of sSet+sSet^+ are the spaces Map♯(X,Y)Map^\sharp(X,Y), which consist of those simplices of the internal homMap(X,Y):=YXMap(X,Y) := Y^X whose edges are all marked:

Remark

When ff is constant on the point, then Nf(C)→N(C)N_f(C) \to N(C) is an isomorphism of simplicial sets, so Nf(C)N_f(C) this is the ordinary nerve of CC.

The fiber of Nf(C)→N(C)N_f(C) \to N(C) over an object c∈Cc \in C is given by taking σ\sigma to be constant on CC. Then all the τ\taus are fixed by the maximal τ(n):Δ[n]→f(c)\tau(n) : \Delta[n] \to f(c). So the fiber of Nf(C)N_f(C) over cc is f(c)f(c).

is not entirely trivial and in fact produces a Quillen auto-equivalence of sSetQuillensSet_{Quillen} with itself that plays a central role in the proof of the corresponding Quillen equivalence over general SS.

Cartesian fibrations over the interval

By the above procedure we can express FF as the image of pp under the straightening functor. However, there is a more immediate way to extract this functor, which we now describe.

First recall the situation for the ordinary Grothendieck construction: given a Grothendieck fibrationK→{0→1}K \to \{0 \to 1\}, we obtain a functor f:K1→K0f : K_1 \to K_0 between the fibers, by choosing for each object d∈K1d \in K_1 a Cartesian morphismed→de_d \to d. Then the universal property of Cartesian morphism yields for every morphism d1→d2d_1 \to d_2 in K1K_1 the unique left vertical filler in

This diagrammatic way of encoding the functor associated to a Grothendieck fibration over the interval generalizes straightforwardly to the quasi-category context.

Definition

Given a Cartesian fibrationp:K→Δ[1]p : K \to \Delta[1] with fibers the quasi-categoriesC:=K0C := K_{0} and D:=K1D := K_{1}, an (∞,1)(\infty,1)-functor associated to the Cartesian fibrationpp is a functor f:D→Cf : D \to C such that there exists a commuting diagram in sSet

and for all d∈Dd \in D, F({d}×{0→1})F(\{d\}\times \{0 \to 1\}) is a Cartesian morphism in KK.

More generally, if we also specify possibly nontrivial equivalences of quasi-categoriesh0:C→≃K0h_0 : C \stackrel{\simeq}{\to} K_{0} and h1:D→≃K1h_1 : D \stackrel{\simeq}{\to} K_{1}, then a functor is associated to KK and this choice of equivalences if the first twoo conditions above are generalized to

Here the left vertical morphism is marked anodyne: it is the smash product of the marked cofibration (monomorphism) Id:D♭→D♭Id : D^\flat \to D^\flat with the marked anodyne morphism Δ[1]#→Δ[0]\Delta[1]^# \to \Delta[0]. By the stability properties discussed at Marked anodyne morphisms, this implies that the morphism itself is marked anodyne.

As discussed there, this means that a lift d:D♭×Δ[1]#→K♯d : D^\flat \times \Delta[1]^# \to K^{\sharp} against the Cartesian fibration in

where the top horizontal morphism picks the 2-horn in KK whose two edges are labeled by ss and s′s', respectively.

Now, the left vertical morphism is still marked anodyne, and hence the lift kk exists, as indicated. Being a morphism of marked simplicial sets, it must map for each d∈Dd \in D the edge {d}×{0→1}\{d\}\times \{0\to 1\} to a Cartesian morphism in KK, and due to the commutativity of the diagram this morphism must be in K0K_0, sitting over {0}\{0\}. But as discussed there, a Cartesian morphism over a point is an equivalence. This means that the restriction

k|D×{0→1}→K0
k|_{D \times \{0 \to 1\}} \to K_0

is an invertible natural transformation between ff and f′f', hence these are equivalent in the functor category.

Conversely, every functor f:D→Cf : D \to C gives rise to a Cartesian fibration that it is associated to, in the above sense.

Proposition

Every (∞,1)(\infty,1)-functor f:D→Cf : D \to C is associated to some Cartesian fibration p:K→Δ[1]p : K \to \Delta[1], and this is unique up to equivalence.

in sSet+sSet^+, where C♯C^\sharp and D♯D^\sharp are CC and DD with precisely the equivalences marked. This comes canonically with a morphism

N→Δ[1]
N \to \Delta[1]

and does have the property that N0=CN_0 = C, N1=DN_1 = D and that ff is associated to it in that the restriction of the canonical morphism D×Δ[1]→KD \times \Delta[1] \to K to the 0-fiber is ff. But it may fail to be a Cartesian fibration.

where the first morphism is marked anodyne and the second has the right lifting property with respect to all marked anodyne morphisms and is hence (since every morphism in Δ[1]#\Delta[1]^# is marked) a Cartesian fibration.

It then remains to check that ff is still associated to this K→Δ[1]#K \to \Delta[1]^#. This is done by observing that in the small object argument KK is built succesively from pushouts of the form

where the morphisms on the left are the generators of marked anodyne morphisms (see here). from this one checks that if the fiber Nα×Δ[1]{0}N_\alpha \times_{\Delta[1]} \{0\} is equivalent to CC, then so is Nα+1×Δ[1]{0}N_{\alpha +1} \times_{\Delta[1]} \{0\} and similarly for DD. By induction, it follows that ff is indeed associated to K→Δ[1]K \to \Delta[1].

To see that the KK obtained this way is unique up to equivalence, consider…